Math 361 Day 4

Random Babies – Inv. B cont’d

Last time – Random Babies simulation

• we mimicked the process of randomly returning 4 babies to their mothers by shuffling and then dealing out 4 slips of colored paper.

• Our class results in part (d) were

Number of matches

0 1 2 3 4

Count 5 14 7 0 1

Proportion 5/27=0.185 14/27=0.519 7/27=0.259 0/27=0 1/27=0.037

Last time – Random Babies simulation • we mimicked the process of randomly returning 4 babies to their mothers by

shuffling and then dealing out 4 slips of colored paper.

• Our class results in part (d) were

• Part (f). The probability of at least one correct match is

0.519+0.259+0+0.037=0.815

According to our simulation of 27 repetitions of the process of randomly returning 4 babies, there is a 81.5% chance that at least one mother will get her own baby.

Number of matches

0 1 2 3 4

Count 5 14 7 0 1

Proportion 5/27=0.185 14/27=0.519 7/27=0.259 0/27=0 1/27=0.037

Inv. B parts (h) and (j)

Number of matches

0 1 2 3 4

Proportion 0.38 0.33 0.249 0 0.041

We can improve our estimates of the probabilities of the numbers of matches by performing more simulations

Inv. B parts (j) and (k)

Number of matches

0 1 2 3 4

Proportion 0.38 0.33 0.249 0 0.041

With 10,000 trials (simulations) of returning 4 babies to their mothers, we estimate the probability of at least one match to be 0.33+0.249+0+0.041=0.62

Random Processes

Definition: An ongoing process whose outcomes have some uncertainty

Example: randomly returning 4 babies to their mothers: this process might return in 0, 1, 2 or 4 correct matches, each with some probability.

Example: tossing a coin: each toss results in “heads” or “tails” with some probability.

Probability

Definition: the probability of a random event is the long-run proportion of times that the event would occur if the random process were repeated over and over under identical conditions.

Example: The probability of a “heads” is 0.5 if a fair coin is repeatedly tossed.

Two ways of analyzing a random process

We can compute the probability of a certain outcome of a random process by either • Simulating the process a large number of times, then

computing the proportion of times the event occurred OR •Assuming a model for the process and using exact

mathematical calculations.

Learning Objectives – Inv. B, Day 4

1. Write out the sample space associated with a random process

2. Compute the value of a random variable for a particular outcome

3. Calculate probabilities using random variables and the assumption of equally likely outcomes.

2. Calculate the expected value of a random variable

Today, we’ll learn how to use exact mathematical calculations to analyze a random process

Some terminology and a principle

Sample space – a list of all possible outcomes of a random process

Random variable – a map between the sample space of a random process and a set of numbers

Principle of equally-likely outcomes – if all n outcomes in the sample space are equally likely to occur, the probability of a particular outcome occurring is 1/n.

Example: coin toss

• Carry out an exact analysis to compute the probability of at least one heads in 3 tosses of a fair coin.

• Compute the expected number of heads in 3 tosses of a fair coin.

1. Write out the sample space associated with a random process

2. Compute the value of a random variable for a particular outcome

3. Calculate probabilities using random variables and the assumption of equally likely outcomes.

2. Calculate the expected value of a random variable

Random Babies - Inv. B cont’d

Generally, we’ll analyze random processes either by simulating the process a large number of times OR by performing exact mathematical calculations. Try parts n, o, p, q, r, s, t and u to see the exact mathematical calculation of the probability of at least one mother receiving the correct baby. Compare with your answer from the simulation (0.62)